Populations and intensities

In Fundamentals E3(b), we saw that the ratio of populations of states of energies E and E is given by 

12 An electronic absorption band consists of many superimposed bands that merge together to give a single broad band with unresolved vibrational structure. 

In condensed media, the width of an electronic transition (that is, the range of wavemunbers, wavelengths, or frequencies over which there is a substantial absorption) results from the simultaneous excitation of molecular vibrations. When a vibration is stimulated by the migration of the electron from one orbital to another, the transition occurs at higher wavenumber than when no vibrational excitation occurs, so the absorption is sUghtly displaced. If several vibrational modes are stimulated, each with a different frequency, and if several vibrational levels of each one are stimulated, then the absorption occurs over a range of frequencies and the electronic transition appears as a broad feature in the spectrum (Fig. 12.12). We treat the excitation of vibration during an electronic transition in more detail in Section 12.6. 

We see that the shorter the lifetime of a state, the less well defined its energy. The energy spread inherent to the states of systems that have finite lifetimes is called lifetime broadening. When we express the energy spread as a wavenumber by writing 6 = hc5v and use the values of the fundamental constants, the practical form of this relation becomes 

Only if T is infinite can the energy of a state be specified exactly (with 6 = 0). However, no excited state has an infinite lifetime therefore, all states are subject to some lifetime broadening, and the shorter the Kfetimes of the states involved in a transition, the broader the spectral hnes. 

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The bandwidth of the laser is limited by an additional intracavity element which is narrow compared with the laser medium bandwidth. The SA and RSA are taken sufficiently thin so that longitudinal variations of the population and intensities can be neglected. The effect of each element of the laser is assumed small enough so that the exponentials can be expanded and truncated to the first two or three orders. With these assumptions an integrodifferential equation for the temporal dependence of the amplitude of the electric field, ACt), can be obtained and the solution to this nonlinear operator equation can be obtained. It takes the form ... 

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